A precise estimation for vibrational energies of diatomic molecules using the improved Rosen–Morse potential

In the context of the generalized fractional derivative, novel solutions to the D-dimensional Schrödinger equation are investigated via the improved Rosen-Morse potential (IRMP). By applying the Pekeris-type approximation to the centrifugal term, the generalized fractional Nikiforov-Uvarov method has been used to derive the analytical formulations of the energy eigenvalues and wave functions in terms of the fractional parameters in D-dimensions. The resulting solutions are employed for a variety of diatomic molecules (DMs), which have numerous uses in many fields of physics. With the use of molecular parameters, the IRMP is utilized to reproduce potential energy curves for numerous DMs. The pure vibrational energy spectra for several DMs are determined using both the fractional and the ordinary forms to demonstrate the effectiveness of the method utilized in this work. As compared to earlier investigations, it has been found that our estimated vibrational energies correspond with the observed Rydberg-Klein-Rees (RKR) data much more closely. Moreover, it is observed that the vibrational energy spectra of different DMs computed in the existence of fractional parameters are superior to those computed in the ordinary case for fitting the observed RKR data. Thus, it may be inferred that fractional order significantly affects the vibrational energy levels of DMs. Both the mean absolute percentage deviation (MAPD) and average absolute deviation (AAD) are evaluated as the goodness of fit indicators. According to the estimated AAD and MAPD outcomes, the IRMP is an appropriate model for simulating the RKR data for all of the DMs under investigation.

where C, B and d are changeable parameters. The Rosen-Morse potential (RMP) was used to explore polyatomic vibrational states of the NH 3 molecule 12 . It was also employed to characterize the diatomic molecular vibrations 13 . By utilizing the equilibrium bond length ( r e ) and the dissociation energy ( D e ) for a DM as explicit parameters, Jia et al. 14 presented an improved expression of the RMP based on the original form of the RMP.

The basics of the GFNU method
The basics of the GFNU method are introduced in this part for solving the generalized fractional differential equation, which takes the following form 49,50 .
where σ (z) and σ (z) are polynomials of maximum 2γ-th degree and τ (z) is a function at most γ-th degree.
Utilizing the primary characteristics of the GFD 42 where D γ [D γ W(z)] =Q 2 z 2(1−γ ) W ′′ (z) + (1 − γ )z 1−2γ W ′ (z) , www.nature.com/scientificreports/ with and inserting Eqs. (5) and (6)  where C ν is a constant of the normalization, and ρ(z) is the weight function given by: The polynomial π GF (z) is determined by: The function h(z) can be obtained if the function under the square root is the square of a polynomial. Hence, the eigenvalue expression is: where Finally, by putting Eqs. (14) and (16) into Eq. (12), the eigenfunctions W(z) can be determined.

Solution of the SE with the IRMP in D-dimensions
The radial SE for a DM in the D-dimensional space with the potential V(r) is given by 50 .
where E, D, J and are the energy eigenvalue, the dimensionality number, and the vibrational quantum number respectively, and is the reduced Planck's constant. By putting, Eq. (21) turns to To determine the approximate analytical solutions of Eq. (25), the Pekeris approximation recipe is applied to the centrifugal term (δ 2 − 1 4 ) r 2 as [19][20][21] where the coefficients b 0 , b 1 and b 2 are defined as follows [19][20][21] Inserting Eq. (26) 3 − 3αr e + 6e −αr e + 3e −2αr e − 2αr e e −αr e + αr e e −2αr e , 18 + 12e αr e + 3e 2αr e − 2αr e e αr e − αr e e 2αr e + 12e −αr e + 3e −2αr e + 2αr e e −αr e + αr e e −2αr e . The negative sign in Eq. (44) is selected to get a physically acceptable solution, the π GF (z) then changes to and Therefore, the functions g(z), τ GF (z) and g ν (z) are written as follows: By integrating Eqs. (47) and (49), the fractional form of the energy eigenvalue of a DM in D dimensions can be expressed as: www.nature.com/scientificreports/ In the absence of the influence of the fractional parameters, the following ordinary expression for the energy eigenvalues can be produced by putting γ = β = 1: By utilizing Eq. (14), the function X(z) becomes Using Eq. (17), the function ρ(z) can be stated as follows With the help of Eq. (16), the function Y ν (z) is written as The complete solution of Eq. (31) is obtained by applying Eq. (12) as follows

Discussion
In this part, the obtained results are applied to a selection of DMs with widespread uses in optical and molecular physics. First, the potential function curves for the chosen DMs are initially generated using the IRMP. The molecular parameters used in this study are presented in Table 1, which are collected from the literature 51-61 . In Figs. (1, 2, 3), potential function curves generated by the IRMP are displayed alongside the experimental RKR points for the considered DMs. These Figs. show that the generated IRMP curves closely correspond to the observed RKR data points [51][52][53][54][55][56][57][58][59][60][61] . We evaluate the average absolute deviations (AAD) from the RKR experimental data in order to demonstrate the effectiveness of the IRMP. A prominent goodness-of-fit metric for evaluating the reliability of an empirical potential energy model is the AAD from the dissociation energy, which is defined as 62 .
where V RKR (r) is the RKR potential and N is the number of experimental data points. Our AAD values for the chosen DMs are shown in Table 2. According to the Lippincott criterion, 62 the AAD of the potential model www.nature.com/scientificreports/ must be less than 1 % of the dissociation energy in order to fit the RKR potential curve. Thus, a better model is indicated by the smaller value of the AAD. As revealed by Table 2, the IRMP is a perfect model for simulating the RKR potential since the computed AAD outcomes for all of the considered DMs are less than 1 % of the dissociation energies. Further potential models for the K 2 (X 1 + g ) molecule that have AAD results are the Morse, Modified Morse, and Hulbert-Hirschfelder potentials 52 . Our AAD value is 0.6999% , whereas the AAD results for the Morse and Hulbert-Hirschfelder potentials are 2.395% , and 0.681% respectively. Consequently, both the IRMP and Hulbert-Hirschfelder potential are superior to the Morse potential for simulating the RKR data of the K 2 (X 1 + g ) molecule. In order to verify the reliability of the expressions generated for the IRMP using the GFNU technique, the pure vibrational energy levels of different DMs are computed in three-dimensional space ( D = 3 ). Comparisons between the calculated energies and the experimental RKR data as well as earlier investigations are provided in Tables 3, 4, 5, 6, 7, 8. To further support the veracity of our findings, we also examine the mean absolute percentage deviation (MAPD) of the IRMP from the RKR experimental points. The MAPD is expressed as 50 : where E RKR are the experimental RKR energies and E nJ are the computed energies using the IRMP. The vibrational energies of the selected DMs are calculated using Eqs. (50) and (53) in both the fractional and ordinary instances respectively. The results in Tables 3, 4, 5, 6, 7, 8 clearly show that the vibrational energies estimated using the IRMP are in close agreement with the RKR experimental data. Also for all of the chosen DMs, the calculated MAPD demonstrates that are within 1% of the allowed error from the experimental RKR values.
The vibrational energies of the ScI ( B 1 ) molecule are displayed in Table 3, along with comparisons to the findings of Refs. [63][64][65] . Diaf et al. employed the path integrals formalism to compute the vibrational energies of the ScI ( B 1 ) molecule with the q-deformed Scarf potential in Ref. 63 . While the modified forms of the generalised Mobius square and hyperbolical-type potentials were used in Refs. 64,65 . The findings of these comparisons show that they coincide with the other potential models [63][64][65] . The vibrational energies for the N 2 (X 1 + g ) molecule are listed in Table 4 compared to the observed RKR data and the outcomes of Refs. 52  www.nature.com/scientificreports/ employed the deformed hyperbolic barrier potential to calculate the energy levels of the N 2 (X 1 + g ) molecule. Whereas the authors of Ref. 52 used the Morse and deformed modified Rosen-Morse (DMRM) potentials. Table 4 illustrates that our findings agree better with the RKR data than those computed using the other potential models 52,66 . Furthermore, our MAPD values are the smallest in both the ordinary and fractional cases. As a result, our IRMP estimates for modelling the N 2 (X 1 + g ) molecule are more accurate than the other works 52, 66 . The vibrational energies for the K 2 (X 1 + g ) molecule are reported in Table 5. When comparing our results with those of Eyube et al. 67 for the K 2 (X 1 + g ) molecule, it becomes clear that our results from the IRMP are more precise for fitting the RKR data for the K 2 (X 1 + g ) molecule than those from the improved q-deformed Scarf oscillator (IQSO) and the ITP. The vibrational energies of the CS(X 1 + ), AsS(X 2 ) and AsP(X 1 + ) molecules are listed in Table 6. As illustrated in Table 6, our outcomes coincide with the RKR data. In Table 7, the computed values for the SrO(X 1 + ), YO(X 2 + ) and ScO(X 2 + ) molecules with the observed RKR values are presented. As can be seen in Table 7, the calculated and observed outcomes are in close agreement. The vibrational energies of the SiP(X 2 ) and SiN(X 2 + ) are listed in Table 8 molecules with the RKR experimental values. It appears that the estimated results and the RKR data agree well. In Table 8, we also provide a comparison of the computed vibrational energies for the SiF + (X 1 + ) molecule with the outcomes of Ref. 31 and observed values. Yanar 31 calculated the vibrational energies for the SiF + (X 1 + ) molecule using the IRMP as well as the improved generalized Pöschl-Teller (IGPT) potential . It is clear that the current findings for the SiF + (X 1 + ) molecule are in good accord with those of Ref. 31 . As illustrated in Tables 3, 4, 5, 6, 7, 8, the influence of incorporating fractional parameters on the vibrational energies for the molecules studied in this work is crucial for modelling the experimental RKR data. Consequently, our results can be investigated to examine various molecules in future studies.

Conclusion
In this paper, the GFD is utilized for the first time to investigate the bound state solutions of the D-dimensional SE using the IRMP. Based on the GFNU, the analytical forms for the energy eigenvalues and wave functions of the IRMP are derived as a function of the fractional parameters in the D-dimensional space by employing the Pekeris-type approximation to the centrifugal term. The present results are applied to a number of DMs that have extensive applications in different physical domains. With the help of the molecular parameters, the potential energy curves are generated in terms of IRMP for the selected DMs. For the chosen DMs, the AAD of the IRMP from the observed RKR data is presented. According to our estimated AAD, the IRMP can successfully fit the experimental RKR data of several DMs. To validate the mechanism used in this research, the pure vibrational energies for different DMs are calculated in both ordinary ( γ = β = 1 ) and fractional ( γ = 1, β = 1 ) cases in three-dimensional space ( D = 3 ). It is found that the current computed pure vibrational energy values are preferable to those from earlier works and are in full harmony with the experimental data. It is further shown that the pure vibrational energies of different DMs computed in the existence of fractional parameters fit the observed RKR data better than those computed in the ordinary case. This leads one to the conclusion that fractional order significantly affects the vibrational energy levels of DMs. The MAPD from the observed RKR data points is assessed to further substantiate the accuracy of our findings. According to the assessed MAPD, our values are accurate to within a 1% error margin of the experimental RKR values. Therefore, the current findings indicate that the IRMP is a precise model for estimating the observed RKR data for all of the DMs considered in this investigation.

Data availability
All data generated or analysed during this study are available upon reasonable request from the corresponding author.